As far as I can tell, rules are scarce on this subject. All we have is:
PHB, Chapter 5, page 144: (on wearing armor, in general)
Armor Proficiency. Anyone can put on a suit of armor or strap a shield
to an arm. Only those proficient in the armor’s use know how to wear
it effectively, however. Your class gives you proficiency with certain
types of armor. If you wear armor that you lack proficiency with, you
have disadvantage on any ability check, saving throw, or attack roll
that involves Strength or Dexterity, and you can’t cast spells.
Summed up: Anyone wearing armor must be proficient, or face the consequences.
PHB, Chapter 5, page 155: (on barding as equipment, price)
Barding. Barding is armor designed to protect an animal’s head, neck,
chest, and body. Any type of armor shown on the Armor table in
this chapter can be purchased as barding. The cost is four times the
equivalent armor made for humanoids, and it weighs twice as much.
Summed up: You can put armor on an animal, and it's price is four times that of normal armor.
PHB, Appendix D, page 310: (variant rule on warhorse specifically)
Variant: Warhorse Armor
An armored warhorse has an Armor Class based on the type of barding worn
(see chapter 5 for more information on barding). Its Armor Class includes the
horse’s Dexterity modifier, where applicable.
Summed up: A warhorse can wear armor. Warhorse, as opposed to Riding horse. Since this is a variant rule, it implies that without the variant rule in effect, all animals (not only warhorses - and war-dogs, war-elephants etc too, I presume) can wear barding.
But do they require proficiency? As per the general rule on armor proficiency quoted at the top; yes. How would an animal go about acquiring such a proficiency? We're not told. Also, we're not told what the difference is between a riding horse and a warhorse, and if/how the first can become the latter.
However, when the RAW is lacking, we just keep rolling. The spirit of the game has always been "make it up as you go". So here is what I would suggest: A non-war-trained mount is not proficient with armor. A war-trained mount is. I believe this is relatively consistent with the rules above.
Also, we need to come up some rules for training a war-animal. Say a month or two of rigorous training in the hands of a skilled person?
Yeah, and just so I've mentioned it: A lawyer might argue that barding and armor are two different things. I'd argue that it's the same thing with different names, depending on the wearer.